Evaluate <mrow> 5 sin &#x2061;<!-- ⁡ 5 x <mo

Alisa Durham

Alisa Durham

Answered question

2022-05-13

Evaluate 5 sin ( 5 x ) 2 ( cos 2 ( 5 x ) + 1 )  d x.

Answer & Explanation

reflam2kfnr

reflam2kfnr

Beginner2022-05-14Added 16 answers

Since 52 is constant with respect to x, move 52 out of the integral.

52sin(5x)cos2(5x)+1dx

Let u2=cos(5x). Then du2=-5sin(5x)dx, so -15du2=sin(5x)dx. Rewrite using u2 and du2.

521u22+11-5du2

Simplify.

52-15(u22+1)du2

Since -1 is constant with respect to u2, move -1 out of the integral.

52(-15(u22+1)du2)

Since 15 is constant with respect to u2, move 15 out of the integral.

52(-(151u22+1du2))

Simplify the expression.

-12112+u22du2

The integral of 112+u22 with respect to u2 is arctan(u2)+C.

-12(arctan(u2)+C)

Simplify.

-12arctan(u2)+C

Replace all occurrences of u2 with cos(5x).

-12arctan(cos(5x))+C

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