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Problem Let I ( x ) = ∫ 3 x x 2 + 1 ln ( 1 + t 2 ) d t I(x)=\displaystyle \int_{3x}^{x^{2}+1}\ln\bigl(1+t^{2}\bigr)\,dt I ( x ) = ∫ 3 x x 2 + 1 ln ( 1 + t 2 ) d t I ′ ( x ) I'(x) I ′ ( x ) 2 x ln ( 1 + ( x 2 + 1 ) 2 ) + 3 ln ( 1 + 9 x 2 ) 2x\,\ln\bigl(1+(x^{2}+1)^{2}\bigr)+3\,\ln\bigl(1+9x^{2}\bigr) 2 x ln ( 1 + ( x 2 + 1 ) 2 ) + 3 ln ( 1 + 9 x 2 ) 4 x x 2 + 1 1 + ( x 2 + 1 ) 2 − 18 x 1 + 9 x 2 4x\,\dfrac{x^{2}+1}{1+(x^{2}+1)^{2}}-\dfrac{18x}{1+9x^{2}} 4 x 1 + ( x 2 + 1 ) 2 x 2 + 1 − 1 + 9 x 2 18 x 2 x ln ( 1 + ( x 2 + 1 ) 2 ) − 3 ln ( 1 + 9 x 2 ) 2x\,\ln\bigl(1+(x^{2}+1)^{2}\bigr)-3\,\ln\bigl(1+9x^{2}\bigr) 2 x ln ( 1 + ( x 2 + 1 ) 2 ) − 3 ln ( 1 + 9 x 2 ) ln ( 1 + ( x 2 + 1 ) 2 ) − ln ( 1 + 9 x 2 ) \ln\bigl(1+(x^{2}+1)^{2}\bigr)-\ln\bigl(1+9x^{2}\bigr) ln ( 1 + ( x 2 + 1 ) 2 ) − ln ( 1 + 9 x 2 ) Browse the problem library