Questions

$$\int \cos x\, e^{\sin x}\,dx$$

$$\int \frac{2ax+b}{ax^2+bx+c}\,dx \quad(a\neq 0)$$

$$\int \cos x\,(1+\sin^2 x)\,dx$$

$$\int \cos x\;\cos^{5}(\sin x)\,dx$$

$$\int \frac{dx}{x\ln x} \qquad(x>0,\ x\neq 1)$$

$$\int \frac{x}{1+x^2}\,dx$$

$$\int \frac{x}{\sqrt{\,1+x^2\,}}\,dx$$

$$\int \frac{e^{x}}{1+e^{x}}\,dx$$

$$\int \frac{\cos x}{\,a+\sin x\,}\,dx \quad(a\in\mathbb{R})$$

$$\int \frac{kx+m}{(kx+m)^2+c}\,dx \quad(k\neq 0)$$

$$\int \frac{\lambda\,(2ax+b)}{ax^2+bx+c}\,dx \quad(a\neq 0)$$

$$\int (3x+1)\,e^{(3x+1)^2}\,dx$$

$$\int_{0}^{\pi/2}\frac{\cos x}{1+\sin x}\,dx$$

$$\int \frac{\ln x}{x}\,dx \qquad(x>0)$$

Solutions

1) Let $u=\sin x$, $du=\cos x\,dx$.

$$\int \cos x\,e^{\sin x}\,dx=\int e^{u}\,du=e^{u}+C=e^{\sin x}+C.$$

2) Let $u=ax^2+bx+c$, $du=(2ax+b)\,dx$.

$$\int \frac{2ax+b}{ax^2+bx+c}\,dx=\int \frac{du}{u}=\ln|u|+C=\ln|ax^2+bx+c|+C.$$

3) Let $u=\sin x$, $du=\cos x\,dx$.

$$\int \cos x(1+\sin^2 x)\,dx=\int (1+u^2)\,du=u+\tfrac{u^3}{3}+C =\sin x+\tfrac{\sin^3 x}{3}+C.$$

4) Let $u=\sin x$, $du=\cos x\,dx$.

$$\int \cos x\,\cos^{5}(\sin x)\,dx=\int \cos^{5}u\,du =\int \cos^{4}u\;\cos u\,du.$$

Write $\cos^{4}u=(1-\sin^{2}u)^2$. Let $w=\sin u$, $dw=\cos u\,du$:

$$\int (1-w^{2})^{2}\,dw =w-\tfrac{2}{3}w^{3}+\tfrac{1}{5}w^{5}+C.$$

Back-substitute $w=\sin u,\ u=\sin x$:

$$\sin(\sin x)-\tfrac{2}{3}\sin^{3}(\sin x)+\tfrac{1}{5}\sin^{5}(\sin x)+C.$$

5) Let $u=\ln x$, $du=\frac{dx}{x}$.

$$\int \frac{dx}{x\ln x}=\int \frac{du}{u}=\ln|u|+C=\ln|\ln x|+C.$$

6) Let $u=1+x^2$, $du=2x\,dx$.

$$\int \frac{x}{1+x^2}\,dx=\tfrac12\int \frac{du}{u} =\tfrac12\ln(1+x^2)+C.$$

7) Let $u=1+x^2$, $du=2x\,dx$.

$$\int \frac{x}{\sqrt{1+x^2}}\,dx=\tfrac12\int u^{-1/2}\,du =\sqrt{1+x^2}+C.$$

8) Let $u=1+e^{x}$, $du=e^{x}\,dx$.

$$\int \frac{e^{x}}{1+e^{x}}\,dx=\int \frac{du}{u}=\ln(1+e^{x})+C.$$

9) Let $u=a+\sin x$, $du=\cos x\,dx$.

$$\int \frac{\cos x}{a+\sin x}\,dx=\int \frac{du}{u}=\ln|a+\sin x|+C.$$

10) Let $u=(kx+m)^2+c$. Then $du=2(kx+m)k\,dx\Rightarrow (kx+m)\,dx=\dfrac{du}{2k}$.

$$\int \frac{kx+m}{(kx+m)^2+c}\,dx =\frac{1}{2k}\int \frac{du}{u} =\frac{1}{2k}\ln\!\big((kx+m)^2+c\big)+C.$$

11) Let $u=ax^2+bx+c$, $du=(2ax+b)\,dx$.

$$\int \frac{\lambda\,(2ax+b)}{ax^2+bx+c}\,dx =\lambda\int \frac{du}{u} =\lambda\ln|ax^2+bx+c|+C.$$

12) Let $u=(3x+1)^2$, $du=6(3x+1)\,dx\Rightarrow (3x+1)\,dx=\tfrac{du}{6}$.

$$\int (3x+1)e^{(3x+1)^2}\,dx=\tfrac16\int e^{u}\,du =\tfrac16 e^{(3x+1)^2}+C.$$

13) Let $u=1+\sin x$, $du=\cos x\,dx$.

$$\int_{0}^{\pi/2}\frac{\cos x}{1+\sin x}\,dx =\int_{u=1}^{u=2}\frac{du}{u} =\ln 2.$$

14) Let $u=\ln x$, $du=\frac{dx}{x}$.

$$\int \frac{\ln x}{x}\,dx=\int u\,du=\tfrac12(\ln x)^2+C.$$