Maths-
General
Easy
Question
The partial fractions of
are
Hint:
In order to integrate a rational function, it is reduced to a proper rational function. The method in which the integrand is expressed as the sum of simpler rational functions is known as decomposition into partial fractions. After splitting the integrand into partial fractions, it is integrated accordingly with the help of traditional integrating techniques.
The stepwise procedure for finding the partial fraction decomposition is explained here::
- Step 1: While decomposing the rational expression into the partial fraction, begin with the proper rational expression.
- Step 2: Now, factor the denominator of the rational expression into the linear factor or in the form of irreducible quadratic factors (Note: Don’t factor the denominators into the complex numbers).
- Step 3: Write down the partial fraction for each factor obtained, with the variables in the numerators, say A and B.
- Step 4: To find the variable values of A and B, multiply the whole equation by the denominator.
- Step 5: Solve for the variables by substituting zero in the factor variable.
- Step 6: Finally, substitute the values of A and B in the partial fractions.
The correct answer is: ![x to the power of 2 end exponent plus fraction numerator 1 over denominator 1 plus 2 x end fraction plus fraction numerator 1 over denominator 1 plus 3 x end fraction](data:image/png;base64,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)
Given :
![fraction numerator 6 x to the power of 4 end exponent plus 5 x to the power of 3 end exponent plus x to the power of 2 end exponent plus 5 x plus 2 over denominator 1 plus 5 x plus 6 x to the power of 2 end exponent end fraction](data:image/png;base64,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)
Step 1: While decomposing the rational expression into the partial fraction, begin with the proper rational expression.
![rightwards double arrow space fraction numerator 6 x to the power of 4 plus 5 x cubed plus x squared plus 5 x plus 2 over denominator 1 plus 5 x plus 6 x squared end fraction space equals space fraction numerator 6 x to the power of 4 plus 5 x cubed plus x squared over denominator 1 plus 5 x plus 6 x squared end fraction space plus space fraction numerator 5 x plus 2 over denominator 1 plus 5 x plus 6 x squared end fraction
F u r t h e r space s i m p l i f y i n g
rightwards double arrow fraction numerator x squared left parenthesis 6 x squared plus 5 x plus 1 right parenthesis over denominator 1 plus 5 x plus 6 x squared end fraction space plus space fraction numerator 5 x plus 2 over denominator left parenthesis 2 x space plus 1 right parenthesis left parenthesis 3 x plus 1 right parenthesis end fraction space rightwards double arrow x squared space plus space fraction numerator 5 x plus 2 over denominator left parenthesis 2 x space plus 1 right parenthesis left parenthesis 3 x plus 1 right parenthesis end fraction](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAfQAAACdCAYAAABCWoqsAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAABOUUD+CQAAEn9JREFUeNrt3X9kHtkawPGjIqqiVERVXCWqqqpKRcWKWqIialWoilirQqxVVbXUtWJda4l11Vp1qVrrqipVtSqiVFVVVKiqWleUWv2j1nWpFRUR5b3nMc+r707PzDvzvnPm1/v9cLTv5H1nzpk5z5yZMzNnjKmnWZsaFclrQ9OWTc9tOlGi/KzYtM+gFxFDxBBQuEMaRI0K5v2ATa9LkpdtNn1l0zOqFDFEDBFDQN522fTEpuGK7oxkB/C2ZHnapFoRQ8QQMQTk7bZNY/r/qu2MBm26YtPZEuXpuE2PqFbEEDFEDAHtjNp0y6Z1PYq9qwHQiUs2fd7y2cfOqBGTspjv5yXaNntM0O06UqH12K1PtEGT+ti8HjtLDBFDJYyhMiO+e9C87nz26+edunIfZ1yJsq6ovkj5v9f1UjTZJku6Q/IV8GUkZ1IzNg3o54O6Q54hhoihksVQ2Rt04ruHyIp8WqJK1ShRRd0sOHBkB7SsO0df6zzvgO9meXttekEMEUM5xRANOvFdOVdTHBXN2bTomP6d/q2sO6NO8j2eciftIz+yIzrgObB8r8esdzCbxBAxlFMM9VqDTnzXwJpJ1xUlj33sbvksN71cqcDZRZJ8N7s1pVLd0yNGU4L8dNLlWuR69BXwcoPYCjFEDOUUQ714hk58V5wEnlxjumPThgluUJCj6qgbFCZtuqz/n7DpQUEVsDlYxLoehV80H67HlDnfvb4eOw347TatmuBmGmKIGMojP3Vp0InvHtvgclQ2ZVOfCZ4dlZX60gQDMbjcN8Hdu3Jn4mAGy2+X4kiej9q0oGdKcd1rZcp3r6zHLNaTPI/9qyl+1DFiqN4xVGfEd4+QI7dhx/QjNr2J+M0FPeo7XIIuolZSIeKeMy1bvntxPabN14gG+z5iiBjKOT91RXzX2LKeUbhsOaY1nx+UbpmZkgR1q6ibKsqW715dj2nyJWcR12zaQQwRQwXkp86I75qSLsEzESt71XE0dVc3gFyHkeuEgwUHdSsZ9/pVxFFgmfLdy+sxab7kxhwZpKWPGCKGCspPXRHfNdav3S9zLSv3mAmeB5xo+d6Qnom0buCTNl0vKKjlBqQxPTOSNKmVdDr0vbLlu9fXY9J8LZnqPG5EDNUzhuqA+O5BQ9r18c6m97pzGg/tsKRiuK4T3tRKkrfTJrjpSPL7Vo/2Rh072jLlm/WYbsdQxiEriSFiqEqIbwAAAABA/XE2icx8bXp3fGIAoEFHbXxj02vDyDpAmh1wt4NgADTo8LYDkiSvKdzG6gIAGnRU189asW54OFAgkaqQynywTapP/cl7XqSa7TuSZmqRM3Qg1x0yQH1BZriGDgA06KgBuct9gNUAADToAACABh1ADXZIkuSNYCuGVxiCOgzqBlBpckOlvDHsGasC1GFQN4Dq22QVgDoM6gZQbcdN8IYwgDoM6ga822/Tgk3PPS6j7M8oy6sP5RWI63rEeVcrazf2mOAa0whVjDpMHaZuUDeQh+s2zXuufGW+c3VeK/h+/Swv55m16XGXO5ElrfSgDlOHqRvUDeSq4fE3jZKW5aBNTzNetlTyZcNb+6jD1GHqBnUDPV7h50ww7G7Yd/q3LJd31aaZjPMllf0A1Yk6TB2mblA3wBFs8CjE7pbPZ2264iFfayZdt1GSfDGWOXWYOkzdoG6g1hVeBiJY1yO8iyZ+iN1Jmy7r/ydseuApX3KDiFwPumPThuZRuqhmPeUL1GHqMHWDuoFSVPhu36zVZ9NRE9x1KkePcV03901wJ6bcnTroKV8NPSqd0rzJYAnyYp6XJhg0odt8gTpMHaZuUDdQqyNYlxMm/hnGC3pEedhjvuSIetgx/YhNbzzkC9Rh6jB1g7qB2lX4ZpeQixxF3tbunxmP+Vo20e+83/KQL1CHqcPUDeoGalfhD9n0yjFdBiqQ5yZ3mOA6lFzzGfSUL+l2OuOYLl1lqx7yBeowdZi6Qd1ApSu83JQxpkeLkia1sk+HvjekR5WtFemkCQaH8JGvfhN0i8ljGX067ZhNL0xwU0jW+QJ1mDpM3aBuwGtF9/0YwmkT3Ijx3qa3JhiGcNRR+SQwXNd8bmqQ+CCV+ZpN7zR/EgDjJcgXqMPUYeoGdQMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAohW027WY1AADEpk2NUNrSxsK3jZyWU7X8uZYbnjZp01tNv9nUV6P1DgBIaZ9Nz3NYxkrE9BclXzcvSrLc8DRpvN/YdEQ/H6xAuQAAHp2y6YbnZUxHLEOWfZNN0NF6mWbdAQBafWvTpQ7+Hp7e/PylTU9N0I1/Sae3duW7fjNo02Wb/muC7uPzjuXJdeLbOt/XNh2PyE94+XG+0Xm9d+QtroySl580v3/YNKV/36XlkDKs2/T30G9lef/Q30j+Htq0J+E6j8pLq59tOuOY3wWbvm8zr6+1XL+YoDs+ajvssOkHm/6nZZBtciXBugYAeHZLzwjT/j08XT5LF/A/bRpJMQ9pTB7b9JkJrukOa0Oxo+V7O03QRfyZfpbu5d8c84paflRjfs2m/hRlb+ZXDhhmTNDtvdem37VhfqQNqkz/m5Zjd8tvX2nDt0vLejniLNu1vlqn3QkdJM3o9C+0sQ03wC/1oCmuXOe1MT8esx36tIyX9P+SLuoB0SlCCQCK9dJ8fEPccOjvwxG/C39vOmIZa9rwueaxZNNAaPrbUAP0Q+hs1+jv+hMu30Wu6U8kWDfhMj7QBrnVGz04cE3f0/Lb8HXunXom3265rmlretDQakob5/CZ/UKCcs078hHeDguhM/3W3+8hlACgOHIm9q7N3zcSTI/6Xrt5vAudATbPAsPz/tOmodC0dfPhLu245Uc5bdOzlrP+pGXcEfG9HTG/j8rfdkeD7vpu0vU9GOq5GNJege1t5uWqA32OZfwRauCT1CEAQA6O6Rln3N8fJpge9b3m3x6lWPaYY94NR1pJuPw4clb5WH870GEZk0yP+s6YYx24vptmfbc2wtKlf6HDeYW3w1FdV0nLDwDIkVx7/aWDv38Rmh43H/nbv1PMOzxdzqBvdVmOOHL2KtfnP28zz6T5dU2P+o5cf15IML+ZFOtbGt0RTa/Mx8+ad1ouuZxxPeN1DwDIiNypPRfz9x/Nx3c6j5qg63Uu4XxkHuciln02QZ7kTHG1y3K0I3eHz7aZp3yej1h2u+musvbrgcSeBPNLmhejB09yEHTDcZASNa+5BN+Tebpu4Lve5boHAGRA7paeivn7t9pAyFnekH6+4vhd3HyuaIOZdNmu6f8xQdex5ENuPJMb5MZTlCPOqM5/sE0+0uQ3PF3+LzeONW/CG9ZpZxPOL836vqDr+0XCbZ60XHIDnzym1xzIRv6VR9bkxr/jhBIAFEvuYt4e83dpPO+ZYBhYuQ5+MuJ3cfM5qDv9hvnrXelRv3FNl+5j6Up+r43vXMpyhP2p+ZH5yfXfAwnykSa/4eny/1PaqMsyn5vox7xc80uzvk9p2T5LOP805ZJ5vtb6II/unTDB8+gMIQsAqL1+bRyzIj0lf8T8XRrdJzmV7ZBhtDoAQI+Qbus7Gc1L7keQ588XY76zbIL7DrImvRj/Mh+u+culihXj7t0AAKB25ImAnzKalzw6Nt3mjHnJUzkGtBzyzPmGHjgcYvMCAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAFCIhqYtm1Zs2tfj+QAAoNK22fSVTc/IBwAA1bcZMV0a2cUS5KNXLeo2AACgreM2PXJMP2zTkxLko9et6LYAACDSHm0wRiIakiOO6aM23bJpXc+o72pj7Csf3WrEpDLYb9OCTc8j/n7UpsdU1cpu509suq3xsqXbeZbNBeKBeMi6IVnSxjTsiDawYfPagO/Xzzt1Yzz2lI+sKnaZXdf1GpfPx9qwo3rbWXqdZmwa0M8HNbZm2GQgHoiHrM7Ml7VBdvnBpnOhabLin+acjywqbKMigRT3u/Mm33sZ2IH5Xd5em16wyUA8EA9x5iJ2/N/p35qkET0QM597No2Hpl1NcRSVVT7ybtCT5jvvBn3MpgdU71ps5yZuAAXxQDy0JY9+7W75fNamK46VH3ddRa5v9IemrZl03eJZ5KOIM/Qk+c67Qe/XbYLqb+fmAdoKmwzEA/HQzqRNl/X/Ex2e2b2POIKS6913bNowwQ0N0gU/6zEfWTXoW9ogSo/ARfPh+o2PfDc8lWeLql2L7bzdplUT3BwEEA/EQ1v3TXDnudxBOJhRg97Qo7gpm/pMMBiMbISXJvpZ6W7z0e5sPs3ZveRZbixb0N6GAxmtv27zRYOerTJv5102/WrTCTYTiAfiIakL2gB0+vyyq8tdpg07vit3xL/xlA9fXTxSgR55Wn90uZdHmbbziO68GN44m9hnyGjioSc0n/OT7pNOHwW47+gCWdaz8qRnj1nkw1eDLjY9rT9uiiuXMmxnOSu6ZtMONkemGDKaeKg1Oeq5qytKrpc8NZ11dbseW5PAOROxcVY95cNXg37Iplee1p+PBv2cbhNUbzvLjUYyEFMfmyP3RgrEQ2UN6Vl064Y4aYKBS9JyDSwj3b7SXTPXsjGOmeD5wQlP+ciiAt3RM9xtmia1Uk97Wn8+GnQGlqnudl4ynT+aifYYMpp4qJ1+3YCua9w3dWOmJeO4H3ZsfOkqeWeCG+ckkMY956Nbp01w457k960eHY7msP7SBELczSQM/Vrt7Vz2oYerzOeQ0cQD8VAreb+cBW68nAX4mO8ho4Ha+dIw5GiR5Lr5vIcjZKDqZ+adDhlNPCBXX2dUUQF2YKjS/iyPIaOJB+TqG5teG0bOAQ06em9/5nvIaOIBXnaqSdL3JvrZb4AdGOq2P/M9ZDTxgEL8rJXtRgaBRKpfynJnm/WOm9RbdSyr/VlTlkNGEw8k7/GQdAaLnKGDMxLU5Aw96f7M55DRxANyxTV0sANDr+7PfA8ZTTwgV3JX6ACrAezA0GP7szyGjCYeANCgR5Ax/us4VsGiiX4tcJ3K3a6ceclryGjigXioQjygpI1nnV/zWPfRBKNG6atbuYsejbCMQ0YTD8QD4FTX1zxKxT/S8rl5/XNdD2LkLuXZEudfhg9d0Hy6RI2jHy73p3p2uWGCN3/JGNo/Gv/dxb7LCeKBeAAi1Ok1j6438snLeuRGpuY104P6nZmSlkG6c2Vo3TRvunOV+6EJ3mLVH/revQqXE8QD8QBEqNtrHmWc+XMJvrfXBK/YzYOPV9eeN3+9Npi03EbPzKpaThAPxEPFteuKyGrll/n1efKKwVta+eSM+q42xt2o42se5Wh7vIueiaTjdhcd2PJO6QcdlFu29ZqncudRThAPxEPFJemKKOpoKg/z2oDv18/yUgm55tXN9ZS6vuZRDnj6E3xvzHzcJdeUZNzuondg/aEzi3bl3q3lkOuGU57KnUc5QTwQDzXR8PibRknLIte3nma87Cxf81g27xN8Z7tNqyZ6UJGsx+1uePrdVoJyh3ubLsTMr9ty51FOEA/EAw16pvPOszvmqkl+s0oRr3ms2g5sl02/2nSizfe6Gbc7q3G5s9iBtZb7lO64j2dU7iLKCeKBeKBBz3zeeXXHrJl03eJ5v+axbOK62kZ055Xkufssx+0uQxdj04DuxHyUmy5G4oF4IB5K06Bv6cqSM9iLJn5oyLy6Y+RGFbneLQNcbGgepQt+1lO+qk6OqF1dh9Ijcc0Ew3e2k/W43XncHBNV7qg65aPc3AREPBAPxEOmK6jbbo8+Ezznt6BnxwcK7o5p6Fn3lOZtm1Y0uZnjqwzyVTeux1Wkx+KWrr92fIzb7SOwz2lZ48rtcljrjo9y51FOEA/EA2foHZHrSo8K7o6RHgPX0JMyIMIbD/mqOteAEksm2T0Dvsbt9hHYSQbSkLp7uuVAUEbK+t2mLzyVO49ygnggHmjQO1Z0d8yyiX5X85aHfNXBk9DBTJJekTKN292u9yZqCMhwucd1572p6aHumExJyt1pOUE8EA806KkdsulVwd0x0q1+xjFdjrBXPeSrDngZRb3LCeKBeKBBjyVHZWN6NrxNj8SkMZ8uuDtGjhilq0geO2te8zpmgmEaJzzkqy6+NPUcMlSun833QLnblRPEA/FQ4Ybc92NWcm1Fbo6Q5xffmuCmkVFH41pEd4w01nJH6jvNnzTw4yXIFwCgBP4PRRK6EVB7L60AAAhSdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1vPiYjeDIxRDI7PC9tbz48bW8+JiN4QTA7PC9tbz48bWZyYWM+PG1yb3c+PG1uPjY8L21uPjxtc3VwPjxtaT54PC9taT48bW4+NDwvbW4+PC9tc3VwPjxtbz4rPC9tbz48bW4+NTwvbW4+PG1zdXA+PG1pPng8L21pPjxtbj4zPC9tbj48L21zdXA+PG1vPis8L21vPjxtc3VwPjxtaT54PC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz4rPC9tbz48bW4+NTwvbW4+PG1pPng8L21pPjxtbz4rPC9tbz48bW4+MjwvbW4+PC9tcm93Pjxtcm93Pjxtbj4xPC9tbj48bW8+KzwvbW8+PG1uPjU8L21uPjxtaT54PC9taT48bW8+KzwvbW8+PG1uPjY8L21uPjxtc3VwPjxtaT54PC9taT48bW4+MjwvbW4+PC9tc3VwPjwvbXJvdz48L21mcmFjPjxtbz4mI3hBMDs8L21vPjxtbz49PC9tbz48bW8+JiN4QTA7PC9tbz48bWZyYWM+PG1yb3c+PG1uPjY8L21uPjxtc3VwPjxtaT54PC9taT48bW4+NDwvbW4+PC9tc3VwPjxtbz4rPC9tbz48bW4+NTwvbW4+PG1zdXA+PG1pPng8L21pPjxtbj4zPC9tbj48L21zdXA+PG1vPis8L21vPjxtc3VwPjxtaT54PC9taT48bW4+MjwvbW4+PC9tc3VwPjwvbXJvdz48bXJvdz48bW4+MTwvbW4+PG1vPis8L21vPjxtbj41PC9tbj48bWk+eDwvbWk+PG1vPis8L21vPjxtbj42PC9tbj48bXN1cD48bWk+eDwvbWk+PG1uPjI8L21uPjwvbXN1cD48L21yb3c+PC9tZnJhYz48bW8+JiN4QTA7PC9tbz48bW8+KzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1mcmFjPjxtcm93Pjxtbj41PC9tbj48bWk+eDwvbWk+PG1vPis8L21vPjxtbj4yPC9tbj48L21yb3c+PG1yb3c+PG1uPjE8L21uPjxtbz4rPC9tbz48bW4+NTwvbW4+PG1pPng8L21pPjxtbz4rPC9tbz48bW4+NjwvbW4+PG1zdXA+PG1pPng8L21pPjxtbj4yPC9tbj48L21zdXA+PC9tcm93PjwvbWZyYWM+PG1zcGFjZSBsaW5lYnJlYWs9Im5ld2xpbmUiLz48bXNwYWNlIGxpbmVicmVhaz0ibmV3bGluZSIvPjxtaT5GPC9taT48bWk+dTwvbWk+PG1pPnI8L21pPjxtaT50PC9taT48bWk+aDwvbWk+PG1pPmU8L21pPjxtaT5yPC9taT48bW8+JiN4QTA7PC9tbz48bWk+czwvbWk+PG1pPmk8L21pPjxtaT5tPC9taT48bWk+cDwvbWk+PG1pPmw8L21pPjxtaT5pPC9taT48bWk+ZjwvbWk+PG1pPnk8L21pPjxtaT5pPC9taT48bWk+bjwvbWk+PG1pPmc8L21pPjxtc3BhY2UgbGluZWJyZWFrPSJuZXdsaW5lIi8+PG1zcGFjZSBsaW5lYnJlYWs9Im5ld2xpbmUiLz48bW8+JiN4MjFEMjs8L21vPjxtZnJhYz48bXJvdz48bXN1cD48bWk+eDwvbWk+PG1uPjI8L21uPjwvbXN1cD48bW8+KDwvbW8+PG1uPjY8L21uPjxtc3VwPjxtaT54PC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz4rPC9tbz48bW4+NTwvbW4+PG1pPng8L21pPjxtbz4rPC9tbz48bW4+MTwvbW4+PG1vPik8L21vPjwvbXJvdz48bXJvdz48bW4+MTwvbW4+PG1vPis8L21vPjxtbj41PC9tbj48bWk+eDwvbWk+PG1vPis8L21vPjxtbj42PC9tbj48bXN1cD48bWk+eDwvbWk+PG1uPjI8L21uPjwvbXN1cD48L21yb3c+PC9tZnJhYz48bW8+JiN4QTA7PC9tbz48bW8+KzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1mcmFjPjxtcm93Pjxtbj41PC9tbj48bWk+eDwvbWk+PG1vPis8L21vPjxtbj4yPC9tbj48L21yb3c+PG1yb3c+PG1vPig8L21vPjxtbj4yPC9tbj48bWk+eDwvbWk+PG1vPiYjeEEwOzwvbW8+PG1vPis8L21vPjxtbj4xPC9tbj48bW8+KTwvbW8+PG1vPig8L21vPjxtbj4zPC9tbj48bWk+eDwvbWk+PG1vPis8L21vPjxtbj4xPC9tbj48bW8+KTwvbW8+PC9tcm93PjwvbWZyYWM+PG1vPiYjeEEwOzwvbW8+PG1vPiYjeDIxRDI7PC9tbz48bXN1cD48bWk+eDwvbWk+PG1uPjI8L21uPjwvbXN1cD48bW8+JiN4QTA7PC9tbz48bW8+KzwvbW8+PG1vPiYjeEEwOzwvbW8+PG1mcmFjPjxtcm93Pjxtbj41PC9tbj48bWk+eDwvbWk+PG1vPis8L21vPjxtbj4yPC9tbj48L21yb3c+PG1yb3c+PG1vPig8L21vPjxtbj4yPC9tbj48bWk+eDwvbWk+PG1vPiYjeEEwOzwvbW8+PG1vPis8L21vPjxtbj4xPC9tbj48bW8+KTwvbW8+PG1vPig8L21vPjxtbj4zPC9tbj48bWk+eDwvbWk+PG1vPis8L21vPjxtbj4xPC9tbj48bW8+KTwvbW8+PC9tcm93PjwvbWZyYWM+PC9tYXRoPin6ZVgAAAAASUVORK5CYII=)
Splitting 5x + 2 in the numerator
![rightwards double arrow x squared space plus space fraction numerator left parenthesis 3 x plus 1 right parenthesis space plus left parenthesis 2 x plus 1 right parenthesis over denominator left parenthesis 2 x space plus 1 right parenthesis left parenthesis 3 x plus 1 right parenthesis end fraction space
S p l i t t i n g space a n d space c a n c e l l i n g space l i k e space f a c t o r s
rightwards double arrow x squared space plus space fraction numerator 1 over denominator left parenthesis 2 x space plus 1 right parenthesis end fraction space plus space fraction numerator 1 over denominator left parenthesis 3 x space plus 1 right parenthesis end fraction space](data:image/png;base64,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)
Thus, the partial fractions of
are ![x squared space plus space fraction numerator 1 over denominator left parenthesis 2 x space plus 1 right parenthesis end fraction space plus space fraction numerator 1 over denominator left parenthesis 3 x space plus 1 right parenthesis end fraction space](data:image/png;base64,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)
Related Questions to study
Maths-
If
then ![P left parenthesis x right parenthesis equals](data:image/png;base64,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)
If
then ![P left parenthesis x right parenthesis equals](data:image/png;base64,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)
Maths-General
Maths-
Let a,b,c such that
then ,
?
Let a,b,c such that
then ,
?
Maths-General
Maths-
If
then B=
If
then B=
Maths-General
Maths-
If the remainders of the polynomial f(x) when divided by
are 2,5 then the remainder of
when divided by
is
If the remainders of the polynomial f(x) when divided by
are 2,5 then the remainder of
when divided by
is
Maths-General
Maths-
Assertion :
can never become positive.
Reason : f(x) = sgn x is always a positive function.
Assertion :
can never become positive.
Reason : f(x) = sgn x is always a positive function.
Maths-General
Maths-
=
=
Maths-General
maths-
Evaluate
dx
Evaluate
dx
maths-General
maths-
If
. The integral of
with respect to
is -
If
. The integral of
with respect to
is -
maths-General
maths-
is
is
maths-General
maths-
dx equals:
dx equals:
maths-General
maths-
If I =
, then I equals:
If I =
, then I equals:
maths-General
maths-
The indefinite integral of
is, for any arbitrary constant -
The indefinite integral of
is, for any arbitrary constant -
maths-General
maths-
If f
= x + 2 then
is equal to
If f
= x + 2 then
is equal to
maths-General
maths-
Let
, then
is equal to:
Let
, then
is equal to:
maths-General
maths-
Statement I : y = f(x) =
, x
R Range of f(x) is [3/4, 1)
Statement II :
.
Statement I : y = f(x) =
, x
R Range of f(x) is [3/4, 1)
Statement II :
.
maths-General