Giải câu 3 bài: Nguyên hàm

Giải câu 3 bài: Nguyên hàm.

a) $\int (1 - x)^{9} d x$

Đặt $u = 1 - x => d u = - d x$

=> $\int (1 - x)^{9} d x = - \int u^{9 d u} = \frac{- u^{10}}{10} + C$

<=> $\int (1 - x)^{9} d x = - \int u^{9 d u} = \frac{- (1 - x)^{10}}{10} + C$

b) $\int x (1 + x^{2})^{\frac{3}{2}} d x$

Đặt $u = 1 + x^{2} => d u = 2 d x$

=> $\int x (1 + x^{2})^{\frac{3}{2}} d x = \frac{1}{2} u^{\frac{3}{2}} d u$

<=> $\int x (1 + x^{2})^{\frac{3}{2}} d x = \frac{\frac{1}{2} u^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} + C$

<=> $\int x (1 + x^{2})^{\frac{3}{2}} d x = \frac{(1 + x^{2})^{\frac{5}{2} + 1}}{5} + C$

c) $\int \cos^{3} x \sin x d x$

Đặt t=\cos x => dt=-\sin xdx$

=> $\int \cos^{3} x \sin x d x = - \int t^{3} d t$

<=> $\int \cos^{3} x \sin x d x = - \frac{t^{4}}{4} + C$

<=> $\int \cos^{3} x \sin x d x = - \frac{\cos^{4} x}{4} + C$

d) $\int \frac{d x}{e^{x} + e^{- x} + 2}$

Đặt $u = e^{x} + 1$

=> $\int \frac{d x}{e^{x} + e^{- x} + 2} = \int \frac{d u}{u^{2}}$

<=> $\int \frac{d x}{e^{x} + e^{- x} + 2} = - \frac{1}{u} + C$

<=> $\int \frac{d x}{e^{x} + e^{- x} + 2} = - \frac{1}{e^{x} + 1} + C$