Topic

Integration by Parts Trigonometric Functions

Topic Progress:

\begin{aligned} \displaystyle \dfrac{d}{dx}\sin{(ax+b)} &= a\cos{(ax+b)} \\ \int{\cos{(ax+b)}}dx &= \dfrac{1}{a}\sin{(ax+b)} \\ \\ \dfrac{d}{dx}\cos{(ax+b)} &= -a\sin{(ax+b)} \\ \int{\sin{(ax+b)}}dx &= -\dfrac{1}{a}\cos{(ax+b)} \\ \end{aligned}

Practice Questions

Question 1

(a)  Find $\displaystyle \int{2x\cos{x}}dx$.

(b)  Hence, find $\displaystyle \int{x^2\sin{x}}dx$.

Question 2

(a)  Find $\displaystyle \int{x\cos{x^2}}dx$.

(b)  Hence, find $\displaystyle \int{x\sin{x^2}}dx$.

(c)  Hence, find $\displaystyle \int{x^3\cos{x^2}}dx$.

Question 3

(a)  Find $\displaystyle \int{\sin^2{x}}dx$.

(b)  Hence, find $\displaystyle \int{x\sin^2{x}}dx$.

Question 4

(a)  Find $\displaystyle \int{\tan^2{x}}dx$.

(b)  Hence, find $\displaystyle \int{\tan{x}}dx$.

(c)  Hence, find $\displaystyle \int_{0}^{\frac{\pi}{4}}{x\tan^2{x}}dx$.

Question 5

(a)  Find $\displaystyle \int{\sec^2{x}\tan{x}}dx$.

(b)  Hence, find $\displaystyle \int_{0}^{\frac{\pi}{4}}{x\sec^2{x}\tan{x}}dx$.

Question 6

(a)  Find $\displaystyle \dfrac{d}{dx}\sec{x}$.

(b)  Hence, find $\displaystyle \dfrac{d}{dx}(\sec{x} + \tan{x})$.

(c)  Hence, find $\displaystyle \int{\sec{x}}dx$.

(d)  Hence, find $\displaystyle \int{x\sec{x}\tan{x}}dx$.

Question 7

Find $\displaystyle \int{\sin{3x}\cos{5x}}dx$.