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Consider the integral âˆ«arctan(4x) dx:

Applying the integration by parts technique, let

u = arctan(4x)

dv/dx = 1

Then, using the formula for integration by parts, we have

âˆ«arctan(4x) dx = uv – âˆ«v du

= xarctan(4x) – âˆ«(1/(1+16x^2))4 dx

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Transcript

for the integral integral arc 10 arc 10 means 10 inverse of 4x into dx. So using the product rule of integration the product rule of integration that is integral of F G dash is equal to F G. This is G minus integral of F dash G. So here let F equal to arc 10 4x arc 10 4x and G dash is equal to 1 so F dash which is equal to d by dx of arc 10 x arc 10 4x and which is equal to the derivative of 10 inverse x is 1 plus 4x square and the derivative of 4x is 4 which is equal to 4 divided by 1 plus 16 x square now G dash is equal to 1 means DG by DX equal to 1 and this further implies DG is equal to DX. Now integrate to both side implies G is equal to X. So substitute the values substitute the values in the formula in the formula product rule formula. So we have integral F G dash is equal to F G minus integral F dash G. So here we have integral of F is arc 10 arc 10 4x and G dash is 1 into DX which is equal to F is F is arc 10 4x into G which is X minus integral of F dash is we already calculated that is 4x 4 divided by 1 plus 16 x square and the G is X into DX. So which…

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